Optimal multipartite entanglement concentration of electron-spin states based on charge detection and projection measurements

We propose an optimal entanglement concentration protocol (ECP) for nonlocal $N$ -electron systems in a partially entangled Greenberger–Horne–Zeilinger (GHZ) pure state, resorting to charge detection and the projection measurement on an additional electron. For each nonlocal $N$ -electron system in a partially entangled GHZ state, one party in quantum communication, say Alice first entangles it with an additional electron, and then, she projects the additional electron into an orthogonal basis for dividing the $N$ -electron systems into two groups. In the first group, the $N$ parties obtain a subset of $N$ -electron systems in a maximally entangled state directly. In the second group, they obtain some less-entangled $N$ -electron systems which are the resource for the entanglement concentration in the next round. By iterating the entanglement concentration process several times, the present ECP has the maximal success probability, the theoretical limit of an ECP as it just equals to the entanglement of the partially entangled state, far higher than others. Moreover, this ECP for an $N$ -electron GHZ-type state requires only one additional electron, not two or more, and it does not resort to a collective unitary evolution, far different from others, which may decrease the difficulty for its implementation in experiment. When it is used for an $N$ -electron W-type state, $N-1$ additional electrons are required only.